Folks,
All this discussion resembles the parable about seven blind
men examining an elephant. Each one examines one part in
detail -- the trunk, the tail, an ear, a side, a leg, the
underside, or the back -- and draws conclusions that are
diametrically opposed to any of the others.
Language has even more sides than an elephant. When we're
doing math, science, or business, we do use the logical side.
But frequently in science and very frequently in business,
we are at a loss about the meaning of some observation or
puzzle, and we have to fall back on vague intuitions.
The hardest part of science is *not* deduction from axioms,
but the *discovery* of axioms that are suitable for precise
deduction. If all we had was logic, we would never be able
to analyze and talk about the typically vague intuitions
that lead to some of the greatest discoveries. But if we
didn't have the ability to do logic, we could never explore
the consequences of those intuitions, and we'd be cheated
in business by any shyster who could.
Following are a couple of quotations:
"As far as the laws of mathematics refer to reality,
they are not certain; and as far as they are certain,
they do not refer to reality." Albert Einstein
"If scientific reasoning were limited to the logical
processes of arithmetic, we should not get very far
in our understanding of the physical world. One might
as well attempt to grasp the game of poker entirely
by the use of the mathematics of probability."
Vannevar Bush
We cannot understand language if we don't recognize that
logic is part of every natural language. But we cannot
use logic effectively unless we recognize that our precise
axioms were derived from some initially vague intuitions.
John Sowa